If it were my yard, I'd say it was Rose of Sharon seedlings; is there a RoS in the area?

You bet! That's was one of my guesses but the leaf shape looked different. Any suggestions for killing/limiting the germination of the RofS seeds?

Just cultivate the area, which will pull them up. They are rooted shallowly at this point, so will come up easily. I do that when the seeds start to germinate, then later pull up the ones I missed. The only ways to avoid this yearly excitement is to grow a sterile RoS or to cut the flowers/branches with flowers off in the fall before they go to seed. I keep saying I am going to do that (prune in fall), but then I remember how many small birds eat the seeds over the winter and I wimp out.

Thanks! This is the first time I am living with a RofS. I will deadhead in the fall and clip the seed pods in a effort to nullify the seedlings.